Theorem syl3anl 1233

Description: A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)

Hypotheses

Ref	Expression
syl3anl.1	⊢ (φ → ψ)
syl3anl.2	⊢ (χ → θ)
syl3anl.3	⊢ (τ → η)
syl3anl.4	⊢ (((ψ ∧ θ ∧ η) ∧ ζ) → σ)

Assertion

Ref	Expression
syl3anl	⊢ (((φ ∧ χ ∧ τ) ∧ ζ) → σ)

Proof of Theorem syl3anl

Step	Hyp	Ref	Expression
1		syl3anl.1	. . 3 ⊢ (φ → ψ)
2		syl3anl.2	. . 3 ⊢ (χ → θ)
3		syl3anl.3	. . 3 ⊢ (τ → η)
4	1, 2, 3	3anim123i 1137	. 2 ⊢ ((φ ∧ χ ∧ τ) → (ψ ∧ θ ∧ η))
5		syl3anl.4	. 2 ⊢ (((ψ ∧ θ ∧ η) ∧ ζ) → σ)
6	4, 5	sylan 457	1 ⊢ (((φ ∧ χ ∧ τ) ∧ ζ) → σ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by: (None)